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Plastic collapse of a stainless steel pressurized tube
Engineering Failure Analysis 17 (2010) 530–536
Contents lists available at ScienceDirect
Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal
Plastic collapse of a stainless steel pressurized tube P. Poza *, C.J. Múnez, R. Rodríguez, J. Rodríguez Departamento de Ciencia e Ingeniería de los Materiales, Universidad Rey Juan Carlos, Escuela Superior de Ciencias Experimentales y Tecnología, C/ Tulipán s.n. 28933 Móstoles, Madrid, Spain
a r t i c l e
i n f o
Article history: Received 28 July 2009 Accepted 7 October 2009 Available online 13 October 2009 Keywords: Stainless steel Plastic collapse Failure analysis Deformation theory of plasticity
a b s t r a c t The failure of a stainless steel tube which conducted oil at 300 °C has been analysed. The fracture surface of the broken tube was studied in the scanning electron microscope and the fracture mechanism found was the nucleation, growth and coalescence of voids. This mechanism is characteristic in materials plastically deformed before failure. Specimens for metallographic examination were cut from the damaged tube and from an intact tube to analyse both microstructures. No significant changes which could justify a microstructure’s embrittlement were detected. Hardness measurements were performed on the damaged and intact tubes. The broken tube was harder than the intact tube due to plastic deformation accumulated during failure. The pressure which is necessary to reach this hardening was analysed by the deformation theory of plasticity and it was found this pressure is close to that corresponding to the plastic instability. Consequently, the most plausible hypothesis of failure was due to an over-pressure which leads to the tube’s plastic collapse. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction A stainless steel tube of 72 mm in diameter and 2 mm thickness, which transports oil at 300 °C failed catastrophically during service. The tube was processed with a DIN 1.4541 stainless steel and was loaded by a nominal internal pressure of 3.6 MPa during service. Several reasons could be considered to explain this failure: an embrittlement process due microstructural changes at high temperature, the presence of critical defects or the plastic collapse due to an over-pressure [1–3]. To analyse all the possibilities the damaged component as well as an intact tube were studied. The damaged and intact materials were characterized, using quantitative metallography, to identify possible microstructural changes which could justify a microstructural embrittlement. In addition, the mechanical properties of the intact and damaged stainless steel were determined to evaluate hardening in the region close to failure. Finally, the fracture surface of the broken tube was analysed to evaluate the dominant fracture mechanisms. 2. Experimental techniques The fracture surface of the broken tube was analysed in a Philips XL 30 scanning electron microscope (SEM) equipped with energy dispersive X-ray microanalysis (EDX). Secondary (SEI) and backscatter electron images (BEI) were obtained. Transverse sections were cut on the as-received failed tube at the rupture edges and at some distance away from the rupture region, in order to examine the microstructural characteristics and differences between the regions adjacent to and * Corresponding author. Tel.: +34 914 887 179; fax: +34 914 888 150. E-mail address: [email protected] (P. Poza). 1350-6307/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfailanal.2009.10.002
P. Poza et al. / Engineering Failure Analysis 17 (2010) 530–536
531
apart from the fractured area. Samples were prepared with standard metallographic procedures [4]. Polished surfaces were etched in Behara’s reagent to reveal the microstructure, and then analysed by light microscopy (LM). HV10 Vickers hardness measurements were performed on both, the failed and intact tubes, using a Vickers hardness tester Wolperttestor Instron 2100. These tests were carried out according to ISO 6507-1 [5]. The purpose is to check if there were indications of different hardness in the analysed areas. Material tensile properties at room temperature and 300 °C were tested on a MTS Alliance RF/100 universal testing machine following the recommendations of UNE-EN-10002 [6]. 3. Failure analysis 3.1. Fractography The fracture surface of the broken tube was analysed by SEM. A low magnification image of the broken surface is presented in Fig. 1 showing a ductile fracture micromechanism [1]. Higher magnification images, along the tube thickness, are presented in Fig. 2. Fracture was initiated at the internal side of the tube and was propagated through the thickness until failure by a mixture of nucleation, growth and coalescence of voids and a tearing mechanism promoted by the small thickness of the tube. These fracture mechanisms are usually observed in materials plastically deformed before fracture. Features which could indicate neither an embrittlement of the stainless steel nor the presence of critical defects were found. Ti (C, N) precipitates were observed through the fracture surface like the one showed in Fig. 3, which are distributed along the material’s microstructure as it will be discussed in the following section. 3.2. Microstructure Normally, under these service conditions, the lost of properties in austenitic stainless steel is associated with intergranular corrosion sensitization, due to preferential precipitation of Cr-rich phases along grain boundaries, usually Cr23C6. Ti additions favour the precipitation of Ti carbides, reducing the risk of sensitization and the appearance of Cr-poor areas in the vicinity of austenite grain boundaries [1,2,7]. The microstructures (observed after etching) of samples drawn from various locations of the failed tube (apart from and adjacent to the fractured area) are illustrated in Fig. 4. In both cases, the microstructure is composed of equiaxed austenitic grains with deformation bands along the conformation direction. In addition, Ti (C, N) inclusions could be detected in both regions. No evidences of secondary phases precipitation at grain boundaries were detected, even at high magnification (Fig. 4b). In the vicinity of the broken zone (Fig. 4c and d) the deformation caused by the fracture was extended about 25–35 lm. However, there is no evidence that the presence of inclusions in the deformed zone was able to encourage the cracking propagation. Therefore, the failure does not seem to be caused by deterioration of the tube’s microstructure. 3.3. Hardness measurements Hardness of the damaged tube, close to the fracture surface, and the intact tube were 240 ± 10 HV10 and 193 ± 6 HV10, respectively. These results show that the damaged tube is 1.24 times harder than the intact one. The analysis of the fracture surface as well as the microstructure of the damaged and intact tubes showed no evidences to justify this hardening by microstructural changes during service. Consequently, this hardening around the damaged section should be explained by plastic deformation accumulated up to failure. To analyse this process it is necessary to determine the hardening curves at room and at the service temperature.
Fig. 1. Fracture surface of the broken tube at low magnification.
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P. Poza et al. / Engineering Failure Analysis 17 (2010) 530–536
Fig. 2. Fracture surface of the broken tube at high magnification showing a ductile fracture mechanism. (a) Internal side of the tube. (b) Middle of the tube thickness (c) External side of the tube.
a)
Ti (C, N)
Intensity
b
Ti
O Si
Cr Ti
Cr
Fe Ni
0
Fe
Al
C N
Ti (C, N)
K
Mg
1
2
3
Ca
4
Ti
5
Cr
Fe
6
7
Ni 8
Energy (keV) Fig. 3. (a) Backscattered electron image showing Ti (C, N) precipitates in the fracture surface. (b) Corresponding EDX microanalysis.
533
P. Poza et al. / Engineering Failure Analysis 17 (2010) 530–536
Fig. 4. Transverse sections of broken stainless steel tube showing an equiaxed austenitic microstructure with precipitates. (a, b) Apart from the fractured area. (c, d) Adjacent to the fractured area.
3.4. Tensile behaviour The mechanical properties of the stainless steel, at room and at the service temperature, are presented in Table 1, where the yield strength at 0.2% offset strain, ry, the ultimate tensile strength, ru, and strain to failure, eu, are included. Fig. 5 shows the hardening curves obtained from the tensile tests. These curves represent the stainless steel’s yield strength as a function of the accumulated plastic deformation. 4. Discussion The fractographic and microstructural analysis presented previously showed no evidences of embrittlement or critical defects which could justify the failure of the component. In addition, the mechanical properties studied shown the broken tube was 1.24 times harder than the intact one. This hardening can be explained by the accumulation of plastic strain before failure and the hypothesis of a plastic collapse is a plausible explanation that should be taken into account. In the following paragraphs, the failure process will be analysed by applying the deformation theory of plasticity [8]. A thin-walled tube is assumed with a ratio between radius and thickness, R/t, of 18. The stress state at any material point in a thin-walled tube subjected to internal pressure, p, can be described by the stress tensor, r, using a cylindrical coordinate system (r, h, z) as:
0 B r¼@
rr 0 0 0
0 rh ¼ pRt 0
1
0 0
rz ¼
pR 2t
C A
ð1Þ
Table 1 Tensile mechanical properties of the steel used to configure the tube at room temperature and at 300 °C. Temperature (°C)
ry (MPa)
ru (MPa)
eu (%)
27 300
430 410
690 470
62.5 20.6
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P. Poza et al. / Engineering Failure Analysis 17 (2010) 530–536
1100 1000
σ (MPa)
900 800 700
Room temperature 300 ºC
600 500 400 0
0.1
0.2
εp
0.3
0.4
0.5
Fig. 5. Stress–plastic strain curves for the steel which configures the tube at room and at the service temperature.
, can be used to define the yielding condition, If the material obeys the Von Mises yield criterion, the equivalent stress, r
r 6 ry :
r ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 3 R 1 p ½ðrr rh Þ2 þ ðrr rz Þ2 þ ðrh rz Þ2 ¼ 2 t 2
ð2Þ
Thus, the maximum admissible pressure, padm, at the service temperature, is:
2 t padm 6 pffiffiffi ry ¼ 26:3 MPa 3R
ð3Þ
This value is clearly higher than the nominal service pressure of 3.6 MPa. Consequently, plastic deformation was induced in the tube by an over-pressure up to this value. Now, the pressure which will induce the component’s plastic collapse will be calculated. The pressure in the tube could be written as a function of the equivalent plastic deformation. Eq. (2) is still valid, but the ratio R/t should take into account the tube’s plastic deformation:
R ¼ R0 expðeh Þ t ¼ t0 expðeh Þ
ð4Þ
R R0 expðeh Þ R0 expðeh er Þ ¼ ¼ t t 0 expðer Þ t0
ð5Þ
where R0 and t0 are the initial radius and thickness, respectively. Plastic deformation is dependent on the loading path. However, under proportional loading histories, such as that of the case here studied, the deformation theory of plasticity can still provide a satisfactory description. Neglecting elastic strains versus plastic strain, the Hencky’s equations propose that the strain tensor, e, is proportional to the deviatoric stress tensor, r0 :
e ¼ w r0
ð6Þ
The deviatoric stress tensor, according to the tube’s stress state, is:
0 pR 2t
0
0
1
B r0 ¼ @ 0
pR 2t
C 0A
0
0
0
ð7Þ
Consequently:
er ¼ eh ¼
PR w; 2t
ez ¼ 0
er eh ¼ 2eh
ð8Þ ð9Þ
From the definition of equivalent plastic strain:
e¼
pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ½ðer eh Þ2 þ ðer ez Þ2 þ ðeh ez Þ2 ¼ pffiffiffi eh 3 3
ð10Þ
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P. Poza et al. / Engineering Failure Analysis 17 (2010) 530–536
Using Eqs. (9) and (10) in Eq. (5):
pffiffiffi R R0 R0 expð2eh Þ ¼ exp 3e ¼ t t0 t0
ð11Þ
Including this value of R/t in Eq. (2), the pressure can be written in terms of the material hardening curve and the equivalent plastic strain as:
pffiffiffi 2 t0 p ¼ pffiffiffi rðeÞ exp 3e 3 R0
ð12Þ
This expression indicates that a plastic instability may appear if the pressure reaches its maximum value. So:
pffiffiffi dr pffiffiffi dp 2 t0 ¼ 0 ¼ pffiffiffi exp 3e 3rðeÞ de de 3 R0
ð13Þ
Thus, the plastic instability condition results:
dr pffiffiffi ¼ 3rðeÞ de
ð14Þ
This condition is showed graphically in Fig. 6. The hardening curve of this steel at 300 °C was determined in the previous (490 MPa), and the equivalent strain, e (0.07), at the instability consection. This graph determines the equivalent stress, r dition, i.e., the tube cannot withstand higher values. This limit could be used in Eq. (12) to obtain the pressure which induces the tube’s plastic collapse:
pffiffiffi 2 t0 p ¼ pffiffiffi rðeÞ exp 3e ¼ 27:8 MPa R 3 0
ð15Þ
The pressure inside the tube should reach this value to promote the failure of the structure by plastic collapse. Any other lower value will be a sign of another type of failure caused by embrittlement or the existence of critical defects. The results obtained in the broken tube would be useful to elucidate this fact. The material analysed from the broken tube was 1.24 times harder than that of the intact one. Considering this hardening only due to accumulated plastic deformation and assuming a proportional relation between hardness and yield stress, the equivalent stress which is necessary to yield the damaged tube at room temperature is also 1.24 times higher than the initial stainless steel’s yield stress. This stress could be introduced in the room temperature hardening curve (Fig. 5) to obtain the plastic strain accumulated in the tube up to failure:
r ¼ 1:24rRT y ¼ 533 MPa
ð16Þ
epaccumulated ¼ 0:053
ð17Þ
These data are referred to room temperature, but to apply this argument to the service temperature an additional hypothesis is needed: the plastic instability is associated with the same amount of accumulated plastic strain. The maximum equivalent stress, which should act over the tube at the service temperature, to reach this critical plastic strain could be obtained introducing Eq. (16) value in the hardening curve at high temperature presented in Fig. 5:
rmaximum ¼ 470 MPa
ð18Þ
1000
1/2
3 σ; dσ/dε (MPa)
900 800
Plastic instability 3 σ=850 MPa σ=490 MPa ε=0.07
700 600 500 400 300 0
1/2
3 σ dσ/dε (MPa)
0.05
0.1
ε
0.15
0.2
p
Fig. 6. Plastic instability condition obtained from the high temperature hardening curve.
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P. Poza et al. / Engineering Failure Analysis 17 (2010) 530–536
From this stress the maximum pressure inside the hardened tube before failure could be obtained substituting the values presented in Eqs. (17) and (18) in Eq. (12):
pffiffiffi 2 t0 p ¼ pffiffiffi rðeÞ exp 3e ¼ 27:5 MPa 3 R0
ð19Þ
This value is very close to the plastic instability condition indicated by Eq. (15). Consequently, the estimated pressure inside the tube before its failure justifies the hypothesis of the plastic collapse. 5. Conclusions The failure of a stainless steel tube conducting oil at 300 °C has been analysed leading to the following conclusions: (a) The fracture surface of the broken tube showed no evidences of embrittlement. Fracture was initiated inside the tube and propagated through the thickness by a ductile mechanism. (b) The microstructure observed in both, the intact and broken tubes, is in agreement with that expected in this kind of steels. Ti (C, N) inclusions were spread through an austenitic matrix. No evidences of microstructural changes during service, which could justify this failure, were found. (c) The damaged tube was 1.24 times harder than the intact one. This hardening should be due to plastic strain accumulated in the tube up to failure, as no relevant microstructural differences were observed. (d) The pressure corresponding to the plastic instability condition is similar to that pressure necessary to justify the tube’s hardening. Consequently, the most plausible hypothesis of failure is the plastic collapse due to an over-pressure.
References [1] [2] [3] [4] [5] [6] [7] [8]
Lamb S. CASTI handbook of stainless steels & nickel alloys, 2nd ed. Materials Parks: CASTI ASM International; 2002. Metals handbook, 9th ed. vol. 13. Metal Park (OH): ASM; 2001. Anderson TL. Fracture mechanics: fundamentals and applications. Boca Raton (USA): CRC Press; 1991. ASTM E407-07. Standard practice for microetching metals and alloys. ISO 6507-1:2005. Metallic materials. Vickers hardness test. Part I: Test method. UNE-EN 10002-1. Metallic materials. Tensile testing. Part I: Method of test at ambient temperature. Jones DA. Principles and prevention of corrosion. Pearson Education, Cop; 1996 Hill R. The mathematical theory of plasticity. Oxford (UK): Oxford University Press; 1950.
Contents lists available at ScienceDirect
Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal
Plastic collapse of a stainless steel pressurized tube P. Poza *, C.J. Múnez, R. Rodríguez, J. Rodríguez Departamento de Ciencia e Ingeniería de los Materiales, Universidad Rey Juan Carlos, Escuela Superior de Ciencias Experimentales y Tecnología, C/ Tulipán s.n. 28933 Móstoles, Madrid, Spain
a r t i c l e
i n f o
Article history: Received 28 July 2009 Accepted 7 October 2009 Available online 13 October 2009 Keywords: Stainless steel Plastic collapse Failure analysis Deformation theory of plasticity
a b s t r a c t The failure of a stainless steel tube which conducted oil at 300 °C has been analysed. The fracture surface of the broken tube was studied in the scanning electron microscope and the fracture mechanism found was the nucleation, growth and coalescence of voids. This mechanism is characteristic in materials plastically deformed before failure. Specimens for metallographic examination were cut from the damaged tube and from an intact tube to analyse both microstructures. No significant changes which could justify a microstructure’s embrittlement were detected. Hardness measurements were performed on the damaged and intact tubes. The broken tube was harder than the intact tube due to plastic deformation accumulated during failure. The pressure which is necessary to reach this hardening was analysed by the deformation theory of plasticity and it was found this pressure is close to that corresponding to the plastic instability. Consequently, the most plausible hypothesis of failure was due to an over-pressure which leads to the tube’s plastic collapse. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction A stainless steel tube of 72 mm in diameter and 2 mm thickness, which transports oil at 300 °C failed catastrophically during service. The tube was processed with a DIN 1.4541 stainless steel and was loaded by a nominal internal pressure of 3.6 MPa during service. Several reasons could be considered to explain this failure: an embrittlement process due microstructural changes at high temperature, the presence of critical defects or the plastic collapse due to an over-pressure [1–3]. To analyse all the possibilities the damaged component as well as an intact tube were studied. The damaged and intact materials were characterized, using quantitative metallography, to identify possible microstructural changes which could justify a microstructural embrittlement. In addition, the mechanical properties of the intact and damaged stainless steel were determined to evaluate hardening in the region close to failure. Finally, the fracture surface of the broken tube was analysed to evaluate the dominant fracture mechanisms. 2. Experimental techniques The fracture surface of the broken tube was analysed in a Philips XL 30 scanning electron microscope (SEM) equipped with energy dispersive X-ray microanalysis (EDX). Secondary (SEI) and backscatter electron images (BEI) were obtained. Transverse sections were cut on the as-received failed tube at the rupture edges and at some distance away from the rupture region, in order to examine the microstructural characteristics and differences between the regions adjacent to and * Corresponding author. Tel.: +34 914 887 179; fax: +34 914 888 150. E-mail address: [email protected] (P. Poza). 1350-6307/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfailanal.2009.10.002
P. Poza et al. / Engineering Failure Analysis 17 (2010) 530–536
531
apart from the fractured area. Samples were prepared with standard metallographic procedures [4]. Polished surfaces were etched in Behara’s reagent to reveal the microstructure, and then analysed by light microscopy (LM). HV10 Vickers hardness measurements were performed on both, the failed and intact tubes, using a Vickers hardness tester Wolperttestor Instron 2100. These tests were carried out according to ISO 6507-1 [5]. The purpose is to check if there were indications of different hardness in the analysed areas. Material tensile properties at room temperature and 300 °C were tested on a MTS Alliance RF/100 universal testing machine following the recommendations of UNE-EN-10002 [6]. 3. Failure analysis 3.1. Fractography The fracture surface of the broken tube was analysed by SEM. A low magnification image of the broken surface is presented in Fig. 1 showing a ductile fracture micromechanism [1]. Higher magnification images, along the tube thickness, are presented in Fig. 2. Fracture was initiated at the internal side of the tube and was propagated through the thickness until failure by a mixture of nucleation, growth and coalescence of voids and a tearing mechanism promoted by the small thickness of the tube. These fracture mechanisms are usually observed in materials plastically deformed before fracture. Features which could indicate neither an embrittlement of the stainless steel nor the presence of critical defects were found. Ti (C, N) precipitates were observed through the fracture surface like the one showed in Fig. 3, which are distributed along the material’s microstructure as it will be discussed in the following section. 3.2. Microstructure Normally, under these service conditions, the lost of properties in austenitic stainless steel is associated with intergranular corrosion sensitization, due to preferential precipitation of Cr-rich phases along grain boundaries, usually Cr23C6. Ti additions favour the precipitation of Ti carbides, reducing the risk of sensitization and the appearance of Cr-poor areas in the vicinity of austenite grain boundaries [1,2,7]. The microstructures (observed after etching) of samples drawn from various locations of the failed tube (apart from and adjacent to the fractured area) are illustrated in Fig. 4. In both cases, the microstructure is composed of equiaxed austenitic grains with deformation bands along the conformation direction. In addition, Ti (C, N) inclusions could be detected in both regions. No evidences of secondary phases precipitation at grain boundaries were detected, even at high magnification (Fig. 4b). In the vicinity of the broken zone (Fig. 4c and d) the deformation caused by the fracture was extended about 25–35 lm. However, there is no evidence that the presence of inclusions in the deformed zone was able to encourage the cracking propagation. Therefore, the failure does not seem to be caused by deterioration of the tube’s microstructure. 3.3. Hardness measurements Hardness of the damaged tube, close to the fracture surface, and the intact tube were 240 ± 10 HV10 and 193 ± 6 HV10, respectively. These results show that the damaged tube is 1.24 times harder than the intact one. The analysis of the fracture surface as well as the microstructure of the damaged and intact tubes showed no evidences to justify this hardening by microstructural changes during service. Consequently, this hardening around the damaged section should be explained by plastic deformation accumulated up to failure. To analyse this process it is necessary to determine the hardening curves at room and at the service temperature.
Fig. 1. Fracture surface of the broken tube at low magnification.
532
P. Poza et al. / Engineering Failure Analysis 17 (2010) 530–536
Fig. 2. Fracture surface of the broken tube at high magnification showing a ductile fracture mechanism. (a) Internal side of the tube. (b) Middle of the tube thickness (c) External side of the tube.
a)
Ti (C, N)
Intensity
b
Ti
O Si
Cr Ti
Cr
Fe Ni
0
Fe
Al
C N
Ti (C, N)
K
Mg
1
2
3
Ca
4
Ti
5
Cr
Fe
6
7
Ni 8
Energy (keV) Fig. 3. (a) Backscattered electron image showing Ti (C, N) precipitates in the fracture surface. (b) Corresponding EDX microanalysis.
533
P. Poza et al. / Engineering Failure Analysis 17 (2010) 530–536
Fig. 4. Transverse sections of broken stainless steel tube showing an equiaxed austenitic microstructure with precipitates. (a, b) Apart from the fractured area. (c, d) Adjacent to the fractured area.
3.4. Tensile behaviour The mechanical properties of the stainless steel, at room and at the service temperature, are presented in Table 1, where the yield strength at 0.2% offset strain, ry, the ultimate tensile strength, ru, and strain to failure, eu, are included. Fig. 5 shows the hardening curves obtained from the tensile tests. These curves represent the stainless steel’s yield strength as a function of the accumulated plastic deformation. 4. Discussion The fractographic and microstructural analysis presented previously showed no evidences of embrittlement or critical defects which could justify the failure of the component. In addition, the mechanical properties studied shown the broken tube was 1.24 times harder than the intact one. This hardening can be explained by the accumulation of plastic strain before failure and the hypothesis of a plastic collapse is a plausible explanation that should be taken into account. In the following paragraphs, the failure process will be analysed by applying the deformation theory of plasticity [8]. A thin-walled tube is assumed with a ratio between radius and thickness, R/t, of 18. The stress state at any material point in a thin-walled tube subjected to internal pressure, p, can be described by the stress tensor, r, using a cylindrical coordinate system (r, h, z) as:
0 B r¼@
rr 0 0 0
0 rh ¼ pRt 0
1
0 0
rz ¼
pR 2t
C A
ð1Þ
Table 1 Tensile mechanical properties of the steel used to configure the tube at room temperature and at 300 °C. Temperature (°C)
ry (MPa)
ru (MPa)
eu (%)
27 300
430 410
690 470
62.5 20.6
534
P. Poza et al. / Engineering Failure Analysis 17 (2010) 530–536
1100 1000
σ (MPa)
900 800 700
Room temperature 300 ºC
600 500 400 0
0.1
0.2
εp
0.3
0.4
0.5
Fig. 5. Stress–plastic strain curves for the steel which configures the tube at room and at the service temperature.
, can be used to define the yielding condition, If the material obeys the Von Mises yield criterion, the equivalent stress, r
r 6 ry :
r ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 3 R 1 p ½ðrr rh Þ2 þ ðrr rz Þ2 þ ðrh rz Þ2 ¼ 2 t 2
ð2Þ
Thus, the maximum admissible pressure, padm, at the service temperature, is:
2 t padm 6 pffiffiffi ry ¼ 26:3 MPa 3R
ð3Þ
This value is clearly higher than the nominal service pressure of 3.6 MPa. Consequently, plastic deformation was induced in the tube by an over-pressure up to this value. Now, the pressure which will induce the component’s plastic collapse will be calculated. The pressure in the tube could be written as a function of the equivalent plastic deformation. Eq. (2) is still valid, but the ratio R/t should take into account the tube’s plastic deformation:
R ¼ R0 expðeh Þ t ¼ t0 expðeh Þ
ð4Þ
R R0 expðeh Þ R0 expðeh er Þ ¼ ¼ t t 0 expðer Þ t0
ð5Þ
where R0 and t0 are the initial radius and thickness, respectively. Plastic deformation is dependent on the loading path. However, under proportional loading histories, such as that of the case here studied, the deformation theory of plasticity can still provide a satisfactory description. Neglecting elastic strains versus plastic strain, the Hencky’s equations propose that the strain tensor, e, is proportional to the deviatoric stress tensor, r0 :
e ¼ w r0
ð6Þ
The deviatoric stress tensor, according to the tube’s stress state, is:
0 pR 2t
0
0
1
B r0 ¼ @ 0
pR 2t
C 0A
0
0
0
ð7Þ
Consequently:
er ¼ eh ¼
PR w; 2t
ez ¼ 0
er eh ¼ 2eh
ð8Þ ð9Þ
From the definition of equivalent plastic strain:
e¼
pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ½ðer eh Þ2 þ ðer ez Þ2 þ ðeh ez Þ2 ¼ pffiffiffi eh 3 3
ð10Þ
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P. Poza et al. / Engineering Failure Analysis 17 (2010) 530–536
Using Eqs. (9) and (10) in Eq. (5):
pffiffiffi R R0 R0 expð2eh Þ ¼ exp 3e ¼ t t0 t0
ð11Þ
Including this value of R/t in Eq. (2), the pressure can be written in terms of the material hardening curve and the equivalent plastic strain as:
pffiffiffi 2 t0 p ¼ pffiffiffi rðeÞ exp 3e 3 R0
ð12Þ
This expression indicates that a plastic instability may appear if the pressure reaches its maximum value. So:
pffiffiffi dr pffiffiffi dp 2 t0 ¼ 0 ¼ pffiffiffi exp 3e 3rðeÞ de de 3 R0
ð13Þ
Thus, the plastic instability condition results:
dr pffiffiffi ¼ 3rðeÞ de
ð14Þ
This condition is showed graphically in Fig. 6. The hardening curve of this steel at 300 °C was determined in the previous (490 MPa), and the equivalent strain, e (0.07), at the instability consection. This graph determines the equivalent stress, r dition, i.e., the tube cannot withstand higher values. This limit could be used in Eq. (12) to obtain the pressure which induces the tube’s plastic collapse:
pffiffiffi 2 t0 p ¼ pffiffiffi rðeÞ exp 3e ¼ 27:8 MPa R 3 0
ð15Þ
The pressure inside the tube should reach this value to promote the failure of the structure by plastic collapse. Any other lower value will be a sign of another type of failure caused by embrittlement or the existence of critical defects. The results obtained in the broken tube would be useful to elucidate this fact. The material analysed from the broken tube was 1.24 times harder than that of the intact one. Considering this hardening only due to accumulated plastic deformation and assuming a proportional relation between hardness and yield stress, the equivalent stress which is necessary to yield the damaged tube at room temperature is also 1.24 times higher than the initial stainless steel’s yield stress. This stress could be introduced in the room temperature hardening curve (Fig. 5) to obtain the plastic strain accumulated in the tube up to failure:
r ¼ 1:24rRT y ¼ 533 MPa
ð16Þ
epaccumulated ¼ 0:053
ð17Þ
These data are referred to room temperature, but to apply this argument to the service temperature an additional hypothesis is needed: the plastic instability is associated with the same amount of accumulated plastic strain. The maximum equivalent stress, which should act over the tube at the service temperature, to reach this critical plastic strain could be obtained introducing Eq. (16) value in the hardening curve at high temperature presented in Fig. 5:
rmaximum ¼ 470 MPa
ð18Þ
1000
1/2
3 σ; dσ/dε (MPa)
900 800
Plastic instability 3 σ=850 MPa σ=490 MPa ε=0.07
700 600 500 400 300 0
1/2
3 σ dσ/dε (MPa)
0.05
0.1
ε
0.15
0.2
p
Fig. 6. Plastic instability condition obtained from the high temperature hardening curve.
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P. Poza et al. / Engineering Failure Analysis 17 (2010) 530–536
From this stress the maximum pressure inside the hardened tube before failure could be obtained substituting the values presented in Eqs. (17) and (18) in Eq. (12):
pffiffiffi 2 t0 p ¼ pffiffiffi rðeÞ exp 3e ¼ 27:5 MPa 3 R0
ð19Þ
This value is very close to the plastic instability condition indicated by Eq. (15). Consequently, the estimated pressure inside the tube before its failure justifies the hypothesis of the plastic collapse. 5. Conclusions The failure of a stainless steel tube conducting oil at 300 °C has been analysed leading to the following conclusions: (a) The fracture surface of the broken tube showed no evidences of embrittlement. Fracture was initiated inside the tube and propagated through the thickness by a ductile mechanism. (b) The microstructure observed in both, the intact and broken tubes, is in agreement with that expected in this kind of steels. Ti (C, N) inclusions were spread through an austenitic matrix. No evidences of microstructural changes during service, which could justify this failure, were found. (c) The damaged tube was 1.24 times harder than the intact one. This hardening should be due to plastic strain accumulated in the tube up to failure, as no relevant microstructural differences were observed. (d) The pressure corresponding to the plastic instability condition is similar to that pressure necessary to justify the tube’s hardening. Consequently, the most plausible hypothesis of failure is the plastic collapse due to an over-pressure.
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